METHODS TO IMPROVE STUDENT ABILITY IN SOLVING MATH WORD PROBLEMS

by

Jennifer Langeness

A Capstone submitted in partial fulfillment of the requirements for the degree of Master of Arts in English as a Second Language

Hamline University

St. Paul, Minnesota

May 2011

Committee: Kathryn Heinze, Primary Advisor Julia Reimer, Secondary Advisor Stacy Natzel, Peer Reader

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TABLE OF CONTENTS

Chapter One: Introduction ................................................................................................. 1

Role of the Researcher ............................................................................................ 4

Background of the Researcher ................................................................................ 5

Guiding Questions .................................................................................................. 6

Summary ................................................................................................................. 6

Chapter Overviews.................................................................................................. 7

Chapter Two: Literature Review ........................................................................................ 8

The Language of Math Word Problems.................................................................. 8

Math Word Problems as a Genre ................................................................ 9

Mathematical Language Difficulties for ELLs ..................................................... 13

Misconceptions ......................................................................................... 14

Background Knowledge............................................................................ 14

Testing....................................................................................................... 15

Student-Authored Word Problems: The Benefits ................................................. 16

Motivation ................................................................................................. 16

Personal Experience .................................................................................. 17

Collaborative Effort .................................................................................. 18

Student-Authored Word Problems: The Implementation ..................................... 18

Direct Instruction ...................................................................................... 19

Scaffolding ................................................................................................ 20

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Cooperative Learning................................................................................ 21

The Gap ................................................................................................................. 22

Research Question ................................................................................................ 23

Summary ............................................................................................................... 23

Chapter Three Overview ....................................................................................... 24

Chapter Three: Methodology ........................................................................................... 25

Overview of the Chapter ....................................................................................... 25

Mixed Method Research Paradigm ....................................................................... 26

Data Collection ..................................................................................................... 26

Location/Setting ........................................................................................ 26

Participants ................................................................................................ 27

Pre- and Post-tests ..................................................................................... 28

Think-Aloud Protocol ............................................................................... 29

Procedure .............................................................................................................. 29

Participants ................................................................................................ 29

Procedural Steps........................................................................................ 30

Materials ................................................................................................... 35

Data Analysis ........................................................................................................ 35

Verification of Data .............................................................................................. 36

Ethics..................................................................................................................... 36

Conclusion ............................................................................................................ 36

Chapter Four: Results ....................................................................................................... 37

Pre-test Results...................................................................................................... 37

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Teacher Journal Results ........................................................................................ 38

Think-aloud Results .............................................................................................. 41

Post-test Results .................................................................................................... 44

Chapter Five Overview ......................................................................................... 48

Chapter Five: Conclusions ................................................................................................ 49

Major Findings ...................................................................................................... 49

Discussion ............................................................................................................. 51

Implications for Teaching ..................................................................................... 53

Limitations of the Study........................................................................................ 54

Professional Growth and Insights ......................................................................... 55

Further Research Recommendations .................................................................... 56

Appendix A: Pre-test questions ........................................................................................ 58

Appendix B: Post-test questions ....................................................................................... 63

Appendix C: Word Problem Procedure (WPP) sheet for students ................................... 68

Appendix D: Think-aloud transcriptions .......................................................................... 70

Appendix E: Parent consent letter – English .................................................................... 78

Appendix F: Parent consent letter – Spanish .................................................................... 80

References ......................................................................................................................... 81

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LIST OF TABLES AND FIGURES

Table 2.1: Syntactic Features of Word Problems ............................................................. 13

Table 3.1: Student Data..................................................................................................... 28

Table 3.2: Example of Word Problem and Corresponding Number Sentence ................. 30

Table 4.1: Student Pre-test Scores .................................................................................... 38

Table 4.2: Student Post-test Scores ................................................................................... 45

Table 4.3: Pre-test and Post-test Comparison ................................................................... 47

Figure 4.1 Pre-test and Post-test Comparison Bar Graph ................................................. 48

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CHAPTER ONE: INTRODUCTION

When one thinks of doing math, initial thoughts may involve working with

numbers and symbols to find answers to problems. Often, math is believed to have less of

a language burden than other content area classes (Chamot & O’Malley, 1994). The

reality is, however, that there is a large amount of language that math students must

attend to in order to understand and solve math problems. This makes learning math for

English Language Learners (ELLs) even more challenging since they are working toward

mastering both basic language proficiency as well as the specialized language needed for

mathematics (Spanos, 1993).

In order for students to do well in the mathematics classroom and on standardized

tests, it is necessary for them to understand the language of math. For non-native speakers

of English, developing the academic language of mathematics is especially challenging.

Much of the work that is done in the classroom and items on standardized math tests are

in the form of math word problems. These problems contain specialized math language

that makes comprehension and solving a challenge for ELLs.

I am an ESL teacher who collaborates with mainstream fourth-grade teachers in

the areas of reading and mathematics. I work with both ESL and mainstream students,

doing whole class instruction and also working with students individually and with small

groups. In the four years that I have been teaching, I have noticed that many ELLs

struggle with comprehending and solving math word problems. In my experience,

students rarely really attempt to understand the word problem. At best, they look for key

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words in the text like “total” and then add numbers together. This strategy is limited,

however, because looking for key words does not always work to the students’

advantage. At worst, I have seen students locate numbers in word problems and perform

a mathematical function with them without knowing why or if what they are doing makes

sense.

As a teacher in the classroom, I have seen ELLs struggle with understanding and

solving math word problems. Many students I have worked with over the years have had

problems comprehending math word problems. They are often unable to translate the

words into number sentences in order to calculate the answers to math word problems. I

have seen them grapple with language that determines meaning in seemingly simple

problems such as How many more apples does Jane have than Peter? This example of

how many more...than is just one of many language related challenges that ELLs face

while learning the academic language of mathematics. In my experience, students have

been confused by the wording of this type of problem and only seem to fully understand

how to solve a how many more...than problem after explicitly being taught that they must

subtract the smaller number from the larger number. This lack of understanding has led

me to search for ways to help students comprehend such problems and solve them with

more accuracy.

In terms of standardized testing, ELLs tend to score lower than their native

English-speaking peers. In fact, a congressionally-mandated study reported that ELLs not

only tend to receive lower grades, but they are also judged by their teachers to have lower

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academic abilities and score below non-ELLs on standardized tests in both reading and

mathematics (Moss & Puma, 1995).

Of all content areas, ELLs struggle most with mathematics and tend to score the

lowest on state and national standardized math tests (Basurto, 1999; Martiniello, 2008).

Standardized tests in math consist primarily of word problems that students must interpret

before they are able to compute answers. It remains unclear whether these problems are

answered incorrectly due to a lack of understanding of the language used or because of

errors in computation. Because of the lack of understanding that ELLs may have of

mathematical word problems based on their limited English proficiency, it is difficult to

know whether their math abilities are being fairly assessed (Martiniello, 2008).

Therefore, for students and schools to be successful in learning and teaching

mathematics, the language of math word problems must be addressed.

Specifically, math language is challenging in word problems because the text is

dense and concept-loaded. Short texts are used to convey larger meaning, which must be

read carefully for accurate comprehension of the problem to be solved (Basurto, 1999).

Some researchers even consider math texts to be a distinct genre for which special

reading comprehension strategies must be taught to ensure accurate understanding

(Winograd & Higgins, 1994/1995).

Research supports that math language is semantically and syntactically

specialized. Math makes use of its own specific and specialized vocabulary such as

denominator and quadrilateral (Rubenstein & Thompson, 2002). Furthermore, students

must learn phrases used in mathematics and learn to distinguish between them. A student

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must know that 6 divided into 12 will yield a completely different result than 6 divided by

12 (Heinze, 2005; Irujo, 2007). Math language also makes use of syntactic features such

as comparative structures, passive voice, and logical connectors that can be a challenge

for ELLs when reading math text (Chamot & O’Malley, 1994).

In addition, ELLs may lack the background knowledge necessary to fully

comprehend math word problems (Barwell, 2001; Short & Spanos, 1989). Word

problems that ask students to calculate the number of fish in a catch or the amount of

change someone would receive with the fewest number of coins assume some cultural

knowledge about fishing and American money. It adds to the complexity of solving math

word problems for ELLs. It is also problematic because children without access to or

experience with this culturally specific information are at a disadvantage when trying to

solve these types of math word problems.

This chapter introduces my plan to scaffold the teaching of students to write their

own word problems as a method of increasing these students’ comprehension of word

problems and their ability to successfully solve them.

Role of the Researcher

I have been an ESL teacher for four years. I work in collaboration with two

fourth- grade mainstream teachers in the areas of reading and math. I have always been

more comfortable teaching reading. The teaching of reading naturally involves teaching

language, which I am most prepared to teach. When I began teaching math, I struggled to

find my role. Language is also a part of math, but how to teach language while teaching

math content wasn’t something that I knew how to do quite as naturally. The one obvious

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area in math where language is especially important is in comprehending math word

problems. I wanted to research how I could help my students grapple with the language

that they need to be successful with these types of problems. This is why I decided to

devote my research to finding out if teaching students a procedure to follow in solving

word problems and helping them to write their own word problems would help them

perform better in this area.

I work with a small group of ELLs during a time allocated for “interventions.” I

thought this would be the perfect chance to help these students become more proficient at

comprehending and solving math word problems. It is for these students that I designed

and carried out my research project.

Background of the Researcher

I have felt frustrated with the limited amount of time that the current math

curriculum spends on developing the strategies necessary to comprehend and solve math

word problems. For that reason, I sought to better understand how to help ELLs

comprehend and solve word problems in their math classes. I wanted to try teaching

students to write their own word problems as a way to help improve their performance.

I believed that students would be motivated to learn how to solve math word

problems by writing their own. Winograd (1992) states that by having students write their

own word problems, they are more receptive to receiving problem-solving instruction

from their teachers. I hoped to encourage my students in the same way.

I used pre- and post-tests in my study in which students were asked to solve word

problems, and later were asked to solve number sentences that corresponded to the

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aforementioned word problems. A bias I had before I began my research was that I

believed that students would be able to solve the number sentences more accurately than

the corresponding word problems. That is to say, I believed that the challenge in solving

math word problems lies in difficulty with comprehending the language more than in the

basic computation of the number sentences.

Guiding Questions

In this paper I will explore the overriding question of how I can help ELLs to

better comprehend and solve math word problems. In terms of my own teaching, I

wanted to know how I could scaffold the process of teaching students to write their own

math word problems to see if this strategy could help them to comprehend word problems

better, to see if this process would help students solve these problems with more

accuracy, and investigate how teaching students to write their own word problems might

help them to talk about how they solve them more proficiently.

Summary

In this chapter, I have focused on the importance of comprehending the language

of math word problems for ELLs. These students are expected to solve math word

problems in order to succeed in their math classes and to do well on standardized tests.

Due to the nature of the language found in word problems, additional strategies may be

needed to help ELLs fully comprehend that language. My aim was to work with students

to help them comprehend and solve math word problems with more accuracy. Working

toward this aim, I scaffolded the process of teaching students to write their own math

word problems to help them comprehend and solve these types of problems more

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accurately. In addition, I strived to help ELLs talk about how they solved math word

problems.

Chapter Overviews

In Chapter One I introduced my research by establishing the purpose, significance

and need for the study. The context of the study was briefly introduced as was the role,

assumptions and biases of the researcher. The background of the researcher was also

given. In Chapter Two I provide a review of the literature relevant to the language of

math word problems and how helping students write their own word problems may help

ELLs to comprehend and solve these word problems better. Chapter Three includes a

description of the research design and methodology that guides this study. Chapter Four

presents the results of this study. In Chapter Five, I reflect on the data collected. I also

discuss the limitations of the study, implications for further research and

recommendations for teachers of ELLs in the content area of math.

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CHAPTER TWO: LITERATURE REVIEW

The purpose of this study is to explore how I can help ELLs to better comprehend

and solve math word problems. In terms of my own teaching, I wanted to know how I

could scaffold the process of teaching students to write their own math word problems to

see if this strategy would help them to comprehend word problems better, to see if this

process would help students solve these problems with more accuracy, and investigate

how teaching students to write their own word problems might help them to talk about

how they solve them more proficiently. Chapter One identified the problem ELLs have

with understanding what math problems mean and then being able to accurately solve

them.

This chapter presents an overview of the specialized language of mathematics and

why this language poses particular difficulties for ELLs. Word problems are challenging

for ELLs both semantically and syntactically. Furthermore, many ELLs lack culturally

specific background knowledge that would assist them in relating to problems. This

chapter also discusses other research that has been done to teach students to write their

own math word problems as a way of increasing their ability to solve such problems.

Finally, the need to use direct teaching and scaffolding to help students author their own

word problems is addressed.

The Language of Math Word Problems

When one thinks about math, symbols and numbers usually come to mind. The

idea that math is very rich in language may not seem apparent at first. The reality,

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however, is that math language is very specialized. In fact, mathematics has its own

distinct language register that makes use of specialized vocabulary and syntax (Kang &

Pham, 1995). Native speakers of English and ELLs alike must learn this special language

of math in order to be successful in mathematics. The language of mathematics is

especially challenging to ELLs, who are learning this academic language alongside the

social language they need for the purpose of basic communication. The complexity lies

primarily in the dense nature of math text, and also in the learning of the semantics and

the syntax of the mathematical register (Kang & Pham, 1995).

Math Word Problems as a Genre

Winograd and Higgins (1994/1995) and also Barwell (2003a) categorize the

language of math word problems as a genre. In Barwell’s article (2003a), he mentions

features that make math word problems a distinct genre. One feature specific to math

word problems is they have a three-part structure, including a “set up” to explain the

scenario of the problem, a number of pieces of information about that situation, and

finally a/some question(s) at the end. Another specific feature mentioned by Barwell is

that one may use the same scenario, but change the information given about it and

therefore be able to formulate word problems that illustrate different mathematical

concepts.

Concept-loaded problems. One characteristic making math text a genre is that

mathematics text is loaded with concepts. Because word problems are short in length,

meaning is densely concentrated into a few words. Unlike other genres of text, word

problems have little context and lack repetitions and rewordings, such as paraphrasing,

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that students usually use to make sense of information in other types of text (Kang &

Pham, 1995). For this reason, students are unable to use the reading comprehension

strategies they have learned to decipher other genres in order to comprehend these word

problems (Irujo, 2007). They must, therefore, learn to read math word problems as a

genre all its own. Because reading comprehension is so crucial to solving math word

problems effectively, instruction in the reading of mathematics texts is important for

ELLs. However, according to Kang and Pham (1995), this instruction is usually not

focused on in the math classroom.

The concept-loaded nature of math text requires the reader to read more carefully,

and at a slower rate than other types of reading (Heinze, 2005) and may require multiple

rereadings (Kang & Pham, 1995). Students who are not aware of how to read

mathematics text may have difficulties. The use of language and concept-loading makes

reading math word problems more difficult for ELLs (Basurto, 1999).

Interestingly, math word problems are often referred to as “story problems.”

While they do give a brief situation, they hardly tell a story. This can be seen in the

following problem:

Sam’s truck weighs 4,725 pounds. The truck can carry 7,500 pounds of rocks.

What is the total weight of the truck and full load?

A real story would give more background information about who Sam is and what his job

is that requires him to haul heavy loads of things in his truck. A real story would

paraphrase and use repetition to make sure that students understand the meaning of the

word “load” in the story (Spanos, 1993).

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Semantics of mathematics. Another characteristic of math as a distinct genre is

semantics. First of all, math makes use of a very specific and specialized vocabulary.

Words like denominator, quadrilateral, parallelogram, and isosceles are special terms

only found in the context of mathematics that math students must learn in order to be

successful (Rubenstein & Thompson, 2002). Lexical items such as divisor and quotient

must be learned in order for students to carry out division problems. Not only are these

words important to learn, but of equal importance are phrases within mathematical

expressions. The phrase divided by gives a completely different equation than the phrase

divided into (Heinze, 2005). For example, 6 divided by 12 is 0.5, whereas 6 divided into

12 is 2 (Irujo, 2007). These subtle yet important distinctions are important yet difficult for

ELLs to recognize.

Also, there are multiple ways to refer to the same function in mathematics. For

example, words referring to the function of addition include sum, add, plus, and,

combine, and increased by. Students must not simply learn one term, but all related terms

to be able to fully comprehend math text. (Chamot & O’Malley, 1994; Heinze, 2005).

Additionally, there is the issue of polysemy, or diversity of meanings, in the

lexicon of mathematics. Words that are used in common speech can also be used for a

specific purpose in mathematics. Students may be familiar with the common uses of

words like quarter, remainder, and place, but these words have a different meaning when

used in mathematics. (Chamot & O’Malley, 1994; Heinze, 2005). Students may know the

various meanings of right in everyday uses such as the right answer or one’s right hand,

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but must assign the word right a different meaning in the mathematical term right angle

(Rubenstein & Thompson, 2002).

Some words used both in math and everyday English have similar meanings, but

the meaning in math is more specific. For example, the word difference in everyday

English denotes a general comparison, while difference used as a math term has the more

precise meaning of the subtraction of two numbers (Rubenstein & Thompson, 2002).

In addition, sometimes the same mathematical word is used in more than one way

within the field of mathematics itself. The word round, for example, can refer to the

shape of a circle or the function of rounding a number to the nearest tenth. Likewise, the

word square can refer to a shape and also to a number times itself (Rubenstein &

Thompson, 2002).

To add to the complexity of math language, some mathematical terms are

homonyms of words used in everyday English. Sum and some, arc and ark, pi and pie are

some examples of words that students may confuse (Rubenstein & Thompson, 2002).

Syntax of mathematics. Syntax is another characteristic making math its own genre. It

is well documented that math language makes use of certain syntactic structures. This is

summed up in Table 2.1. In addition to the syntactic features listed in the table, other

features that can be problematic include relative and subordinate clauses (Short &

Spanos, 1989). Pronouns used to refer back to referents (Heinze, 2005) can also be

challenging if learners have not mastered the English pronoun system. For example, in

the following problem where them is used to refer back to toy cars and she is used to

refer back to Rachel.

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Rachel had 17 toy cars. She gave 11 of them away. How many toy cars does she

have now?

Table 2.1

Syntactic Features of Word Problems

1. Comparatives 2. Prepositions 3. Passive voice 4. Reversals 5. Logical connectors

greater/less than n times as much as as…as divided into divided by by if…then

6 is greater than 4 Maria earns six times as much as Peter. Lin is as old as Roberto. 4 (divided) into 8 10 divided by 5 2 mulitplied by 6 (x) vs. x exceeds 2 by 7 (+) x is defined as a number greater than 7 When 5 is added to a number, the result is 7. The number a is five less than b. Correct equation: a = b – 5 or b – a = 5 Incorrect equation: a = 5 – b or a – 5 = b If a is positive then –a is negative.

Note. From from CALLA Handbook: Implementing the cognitive academic language learning approach, p. 230 by A. Chamot and J.M. O’Malley, 1994, MA: Addison-Wesley Publishing Company.

Mathematical Language Difficulties for ELLs

While math word problems are a source of difficulty for non-native English

speakers and native English speakers alike, there are reasons that such problems are

especially challenging for students whose native language is not English. These reasons

include teacher misconceptions about the language burden of mathematics and a lack of

culturally specific background knowledge. These barriers often result in non-native

14

English speakers scoring much lower on standardized math tests than their native

English-speaking peers (Moss & Puma, 1995).

Misconceptions

Research indicates that teachers believe that mathematics does not involve as

much language as that of other content areas (Chamot & O’Malley, 1994). Other authors

address the claim that mainstream teachers believe that ELLs do not need to learn as

much language as they need to learn math symbols and how to manipulate these symbols

(Kang & Pham, 1995). As has been supported in the previous section, this is simply not

true. The language demands of mathematics can be quite challenging.

The truth is that there is a heavy language burden in mathematics. Students

learning English do not need less language in the mathematics classroom, but rather more

(Kang & Pham, 1995). Students who are working toward English proficiency need to

master both basic language proficiency as well as the special register necessary to be

successful in mathematics (Spanos, 1993). Because of this misunderstanding of the

language burden for ELLs, many teachers limit the teaching of math to computation

exercises instead of problem solving efforts in English because these teachers believe that

ELLs cannot be successful in solving word problems until they are more fluent in the

English language (Basurto, 1999).

Background Knowledge

Another reason ELLs might find math word problems difficult is because they

have a lack of background knowledge about specific problems (Barwell, 2001; Short &

Spanos, 1989). Many math word problems contain culturally specific information.

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Students who are not familiar with this information will struggle with understanding and

solving these problems. For example, on Minnesota’s Sample Fourth-grade MCA test

(Minnesota Department of Education, 2009), one question reads:

Oliver went fishing. He caught 2 pike, 1 muskie, and 3 bass. What fraction of

Oliver’s total catch was bass?

Students who lack background knowledge of fishing may find this question confusing.

They may be unfamiliar with the names of fish and the term catch because here it is used

as a noun to mean the group of fish Oliver caught, instead of as a verb, which is its more

common meaning.

Students with little prior knowledge about American money may have difficulty

answering questions like the following, also from Minnesota’s Sample Fourth-grade test

(Minnesota Department of Education, 2009).

Maria gave a store clerk $10.00 for a purchase that cost $9.19. How would Maria

receive her change using the least number of coins possible?

Without the knowledge of U.S. coins, students would not be able to successfully answer

this question.

Testing

In terms of standardized testing, ELLs tend to score lower than their native

English-speaking peers. Of all content areas, ELLs struggle most with mathematics and

tend to score the lowest on state and national standardized math tests (Basurto, 1999;

Martiniello, 2008). Because of the lack of understanding that ELLs may have of

16

mathematical word problems based on their limited English proficiency, it is difficult to

know whether their math abilities are being fairly assessed (Martiniello, 2008).

Student-Authored Word Problems: The Benefits

Several researchers have advocated for students writing their own math word

problems as a way to increase their ability to comprehend and solve such problems, as

well as to increase problem solving ability in general (Barwell, 2001, 2003b; Heinze,

2005; Martiniello, 2008; Short & Spanos, 1989; Spanos, 1993; Winograd, 1990, 1992;

Winograd & Higgins, 1994/1995). Furthermore, it can have a positive effect on student

motivation and interest, it links math to students’ personal experience and background

knowledge, and allows students and teachers to work collaboratively to create the math

curriculum. This is particularly beneficial to ELLs. Linking content to personal

experience and background knowledge is especially crucial for non-native speakers of

English who are trying to navigate the language of math word problems.

Motivation

One reason for learners to write their own math word problems relates to interest

and motivation. Students tend to get excited when word problems are about them or their

classmates. They have a natural curiosity that can be tapped into during the mathematics

class by making the content of word problems more relevant (Winograd, 1995). This

curiosity lends itself well to students creating their own word problems. It channels

students’ wonderings into creating situations of their own. Students who write their own

math word problems tend to be more interested in the problem solving process

(Winograd, 1995). This leads to a captive audience for teachers, who can provide direct

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problem solving instruction to students who are more receptive. Winograd (1992)

conducted a case study with mainstream fifth grade students. He observed one class as

they wrote, solved, and shared math word problems in small groups. He found that

students are more willing to accept problem solving instruction from teachers if that

instruction is directly related to solving their self-authored word problems. He also

discovered that students tend to write word problems that require more complicated skills

to solve than they currently possess. For example, one student wrote a word problem that

required her to find the volume of her recorder, the musical instrument the students were

learning to play. This created an opportunity for the teacher to teach that student how to

solve for volume in order for the student to solve her self-authored word problem.

Personal Experience

Because context is important for the comprehension and solving of math word

problems (Chapman, 2006), students who have had personal experience with the

situations described in the word problems they read will have more success

understanding and solving them. When students write their own problems, they make use

of contexts and situations with which they are intimately familiar. This helps to alleviate

the problem of students having to solve word problems that are culturally unfamiliar to

them. It incorporates their background experience so that math word problems are more

comprehensible to students.

Generally, when students author math word problems, they tend to include

information about their actual experience, their hobbies, information from other content

areas, objects in the classroom environment, or they write about topics that come from

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their imaginations (Winograd, 1992). Problems written by students are more authentic,

because they reflect the background knowledge and personal experience that students

bring from their own lives. The use of personal experience in creating math word

problems can make problems more meaningful to students (Barwell, 2003b).

Collaborative Effort

Often times, students are asked to solve only problems they see in their textbook

or that have been written by their teacher. By having students write their own math word

problems, they are essentially constructing the math curriculum together as a class and

with the teacher (Winograd, 1992). Students can share their math problems with the class

as a whole group by writing the word problems they created on the board and inviting

their peers to solve them. Teachers can also publish student-written word problems on

worksheets and ask students to solve their classmates’ problems. In this way, students are

not only solving problems that adults create for them, but they are also able to solve

problems that they create for each other (Winograd, 1992). This is particularly powerful

for ELLs.

Student-Authored Word Problems: The Implementation

Teaching students to write their own word problems can be done in a few

different ways in the classroom. Generally, there should be direct problem solving

instruction from teachers, teacher scaffolding of the problem writing process, and a

component of cooperative learning for students to share and solve their problems.

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Direct Instruction

Much of the research stresses the need for direct problem-solving instruction as

students learn to write their own math word problems. Students need to learn a procedure

for solving problems (Chamot & O’Malley, 1994; Heinze, 2005; Spanos, 1993;

Winograd, 1990). Spanos (1993) has devised an eleven step procedure to help guide

students through the problem solving process that he calls Word Problem Procedure

(WPP). The following is his list of steps:

1. Choose a partner or partners.

2. Choose a problem. Write the problem in the space below.

3. One student read the problem out loud. Discuss the vocabulary and circle words you

don’t understand. Write the words below.

4. Use a dictionary for help. Ask your partner or teacher for help.

5. What does the problem ask you to find? Write this below.

6. What should you do to solve the problem? Add? Subtract? Multiply? Divide? Write

this below.

7. Solve the problem below.

8. Check your answer below.

9. Explain your answer to your partner(s). Write your explanation below.

10. Explain your answer to the class.

11. Write a similar problem on the back of this page.

When students follow the steps of this procedure, they are able to practice the

mathematical academic English in the four modalities: reading, writing, listening, and

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speaking. Students are also using math reasoning strategies, applying math concepts, and

using social strategies as they work collaboratively (Spanos, 1993).

This procedure is especially beneficial for students with limited English

proficiency because it is a means to breaking down and understanding word problems.

Spanos (1993) conducted a case study in a high-school ESL math class. When taught the

word problem procedure, the students in Spanos’ study made progress in solving math

word problems even though they had trouble explaining their solutions to such problems.

Also, by writing similar word problems, these students were able to focus on the word

problem structure and understand in mathematical terms what was required of them.

Scaffolding

Students also need scaffolding to help them write their word problems (Heinze,

2005; Spanos, 1993; Winograd, 1990; Winograd & Higgins, 1994/1995). At the

beginning, teachers can model the process showing students their own teacher-generated

word problems created from their own experience. Teachers can discuss how they chose

their topics and teachers and students can discuss how these problems could be solved.

As students work on solving these teacher written problems, the teacher will provide

direct instruction of steps (such as Spanos’ WPP) to help students solve the problems.

Teachers can then begin to provide students with word problems that contain missing

information. Students first provide simple information as they fill in blanks. As time goes

on, students provide more information and eventually write an entire word problem on

their own.

21

For example, students could be given this word problem used in Spanos’ 1993

study:

Sam’s truck weighs 4,725 pounds. The truck can carry 7,500 pounds of rocks.

What is the total weight of the truck and full load?

Students could solve the above problem and then write a similar problem by filling in

information and/or numbers into the blanks like this:

Sam’s _______ weighs _______ pounds. The _______ can carry _______ pounds

of rocks. What is the total weight of the _______ and the full load?

In this way, teachers can scaffold the process by which students write their own word

problems. With time, less information could be provided to students until they are able to

write entire problems on their own. For example, students can make up problems using

different names and vehicle types, and also different loads and weights that the vehicle

carries.

Scaffolding the teaching of word problems is an important step. Using old

problems to solve new ones helps deepen students’ mathematical understanding and

helps to integrate the new and old knowledge (Winograd & Higgins, 1994/1995).

Cooperative Learning

Most of the literature stresses the benefits of students sharing their original

problems with their peers (Chamot, Dale, O’Malley, & Spanos, 1992; Chamot &

O’Malley, 1994; Hildebrand, Ludeman, & Mullin, 1999; Winograd, 1990). This allows

for discussion that will deepen the understanding of math word problems. In fact, Chamot

22

and O’Malley (1994) claim that when students work together cooperatively to discuss

their problems, it is more effective than if they work alone or in pairs.

In some models, students share the problems they write with small groups of

students. In other models, students take turns presenting their problems to the whole class

(Chamot, et al., 1992; Chamot & O’Malley, 1994; Hildebrand, et al., 1999; Winograd,

1990). In all cases, it is an opportunity for students to hear feedback about the problems

they create. It leads small groups or whole class negotiation of meaning about the

problem and how best to solve it. The discussion also gives students practice talking

about math and using math language.

The Gap

Clearly, the language burden is great for ELLs in the area of math. With such an

emphasis placed on standardized math testing that involves the language of math word

problems, teachers need to concentrate on making sure ELLs are able to comprehend and

solve such problems. The research suggests that scaffolding the teaching of students to

author their own word problems can help students learn to write them. Some benefits of

this instruction and some positive results have been reported. More research is needed,

however, to find out if ELLs will benefit from learning to write their own math word

problems. There has been little research done in this area, and much of the existing

research that was done by Winograd (1990, 1992), Winograd & Higgins (1994/1995),

and Spanos (1989, 1993) is now a bit dated. I wish to address this gap in the research.

23

Research Question

Through my research, I hoped to learn the value of scaffolding the teaching of

ELL students to write their own math word problems. Therefore my questions included:

Would scaffolding the teaching of students to write their own math word problems help

them comprehend these types of problems better? Would this process make the language

of math more comprehensible? At the end of my instruction, would students be able to

solve math word problems more accurately? Finally, would students be able to talk about

how they solved math word problems with greater ease?

Summary

In this chapter, I have presented information about the specialized language of

mathematics and why this language poses particular difficulties for ELLs. Mathematics

language is challenging because it has its own distinct semantic and syntactic structure.

Word problems in math tend to be highly dense and difficult for ELLs to unpack.

Misconceptions that teachers have about the importance of math language and a lack of

culturally specific background knowledge required to solve word problems make matters

even more challenging for ELLs.

I have also included in this chapter a look at other research that has been done to

teach students to write their own math word problems as a way of increasing their ability

to solve such problems. Benefits to ELLs in writing their own math word problems

include increased motivation and interest, a greater link to personal experience and

existing background knowledge, and collaboration between students and teachers to

collectively write the math curriculum.

24

Chapter Three Overview

Chapter Three gives a detailed account of the methods used to conduct my

research. It also includes information about the participants, the setting, and techniques

for data collection and analysis.

25

CHAPTER THREE: METHODOLOGY

This study was designed to explore how to help ELLs better comprehend and

solve mathematical word problems. In this study, I wanted to find out how teachers could

scaffold the process of teaching students to write their own math word problems, to see if

teaching students to write math word problems would help them to comprehend word

problems better, and also to learn if this process would help students be able to solve

math word problems with more accuracy. Lastly, I wanted to investigate if teaching

students to write their own word problems would help them to talk about how they solve

math word problems.

First, I gave a pre-test using selected word problems from the Sample Fourth-

Grade MCA Math Test (Minnesota Department of Education, 2009). To find out if

teaching students to write their own word problems helps students comprehend word

problems and talk about how they solve a problem, I used tape-recorded think-aloud

verbal reports as students solved a word problem. Finally, in order to discover if teaching

students to write their own word problems helped students to solve word problems with

more accuracy, I conducted a post-test using different selected word problems from the

Sample Fourth-Grade MCA Math Test (Minnesota Department of Education, 2009).

Overview of the Chapter

In this chapter I will describe my use of a mixed method research paradigm. In

addition, I will give information about the participants, the setting, and my techniques for

data collection and analysis.

26

Mixed Method Research Paradigm

I used both qualitative and quantitative research to explore my research questions.

According to Dörnyei (2007), this combination of qualitative and quantitative research is

referred to as “mixed methods research.” It allows the researcher to combine the rich,

deep, and descriptive data of qualitative research with the more objective, number-driven

data of quantitative research. I think that using both methods gave me more well-rounded

data and that my findings were more valid.

The pre- and post-tests I administered were quantitative. The pre-test score was

compared with the post-test score to determine if progress had been made in

comprehending and solving math word problems. Qualitative research was conducted by

administering think-aloud reports. This data allowed me access to student thinking as

they solved math word problems. I also kept a teacher journal that provided me with

additional insights.

Data Collection

Location/Setting

This study was completed in a small urban Midwest elementary school. The student

population is around 450 students. Sixty-one percent of the students receive free or

reduced lunch. About twenty-five percent of the student population is Limited English

Proficient. The total minority population at the school is fifty-five percent. The school

uses a collaborative model for ESL instruction. As the ESL teacher, I work within the

two fourth-grade classrooms where ELLs are clustered. I work collaboratively with the

27

mainstream teacher during Reader’s Workshop and math to ensure that the curriculum is

accessible to ELLs.

Data was collected during a daily thirty-minute small group designed as an

intervention for both reading and math. Students received instruction in reading four days

during a six-day rotation and in math two days per six-day rotation. ELLs were chosen to

participate in this group based on Measures of Academic Progress (MAP) scores from the

fall of the 2010-11 school year that show they do not meet or partially meet state

standards in reading and/or math.

Participants

The participants in this study were six fourth-grade ELLs. The level of English

proficiency of the students is intermediate based on the Test of Emerging Academic

English (TEAE) scores in reading and writing and also Student Oral Language

Observation Matrix (SOLOM) evaluations.

The TEAE is an exam that ELLs are required to take each year. It consists of two

parts; one to assess reading ability and the other to assess writing ability. The SOLOM is

a rubric completed by ESL teachers that assesses listening and speaking abilities. The six

students I have chosen are primarily Spanish-speaking and have been educated in the

U.S. school system from one and a half to five years. Table 3.1 gives more specific

information about these students.

28

Table 3.1

Student Data

Student Home Language

Amount of Time in U.S.

Schools

Fall 2010 MAP Math Score

SOLOM Score 2010

TEAE Score 2010

1 Tibetan 5 years 206 4 Reading, 2 Writing, 3

2 Arabic 1.5 years Not available Not available Not available

3 Spanish 5 years 201 3 Reading, 3 Writing, 4

4 Spanish 4 years 193 3 Not available

5 Spanish 5 years 212 3 Reading, 3 Writing, 4

6 Spanish 4 years 194 3 Reading, 3 Writing, 4

Pre- and Post-tests

To begin with, I needed to find out how well students solved word problems to

get a baseline of their proficiency. Therefore, my first data collection method was a pre-

test using MCA-II math word problems from the fourth-grade sample test posted online

by the MN Department of Education (Minnesota Department of Education, 2009). To

discover if students had improved in their ability to solve word problems, I gave them a

post-test, with different math word problems from the same MCA-II fourth-grade sample

test. The scores from the two tests were compared to see if students improved their

scores.

29

Think-Aloud Protocol McKay (2006) outlines the use of the think-aloud as a way of gaining access to

students’ thinking as they complete a task. Students are asked to complete a task, which

in my study was reading and solving a math word problem. As students are working on

their task, they are prompted by the teacher to verbalize what they are thinking. I

recorded students as they solved a math word problem, and then transcribed the sessions

for later analysis. My reason for conducting think-alouds with students was to see if there

was greater understanding in the comprehension of math word problems. Learning what

they comprehend or do not comprehend of a word problem can only be accomplished by

listening to the thoughts of students as they solve problems.

Procedure

Participants

ELL students were chosen to participate in this group based on MAP scores from

the fall of 2010 that show they only partially met state standards in reading and/or math.

The purpose of the group instruction was to help these students become more proficient

in reading and math to eventually meet the state standards. The participants were first

given a pre-test to determine a baseline of ability to solve math word problems.

Following the pre-test was ten weeks of instruction focused on learning a Word Problem

Procedure for solving word problems. As part of the procedure, students learned through

a scaffolded process to write their own word problems. At the end of the ten-week

instruction period, participants individually participated in a think-aloud, where they

30

solved a math word problem while explaining how they were solving it. Finally, a post-

test was administered to check for improvement in solving math word problems.

Procedural Steps

Pre-test. First, I gave participants a pre-test using five selected word problems taken

from the Sample Fourth-Grade MCA Math Test (Minnesota Department of Education,

2009). The participants completed this written pre-test during the small group

intervention time. After collecting students’ papers, I then gave them these same five

problems written in number sentence form in a different order on a separate paper. See

Appendix A for the pre-test.

After correcting the two written parts of the pre-test, I compared each word

problem with its number sentence counterpart for each student and recorded scores to

reflect the number of word problems correctly answered and the number of number

sentence problems correctly answered. The five word problems I selected represented at

least one addition, one subtraction, one multiplication, and one division problem to assess

a variety of mathematical functions.

Table 3.2

Example of word problem and corresponding number sentence

Example word problem:

Last year a basketball player scored 513 points. This year he has scored 466 points. How

many more points must he score to have the same score as last year?

Corresponding number sentence:

513 – 466 = ___________

31

Test items were framed in both formats so that I could see which format students

struggled with the most. My assumption was that students would answer the word

problems with less accuracy and the number sentence problems with more accuracy. By

comparing the two formats, I was able to see if students were struggling with the

language in the word problem, computation, or both.

Instruction. I instructed the group of participants in a small group setting over a ten-

week period. The small group met daily, but focused on math an average of one day per

week for 30 minutes each session. In all, the group received a total of 12 sessions of

instruction totaling six hours focused on solving word problems.

I began by teaching students a revised version of Spanos’ Word Problem

Procedure (WPP) (Spanos, 1993). The steps of this procedure I used with the students

were the following:

1. Read the problem out loud.

2. Talk about the vocabulary and circle words you don’t understand. Write the words

below.

3. Ask your partner or teacher for help with what these words mean.

4. What does the problem ask you to find? Write it below. (What question does it ask

you?)

5. Draw a picture to represent the problem.

6. What should you do to solve the problem? Add? Subtract? Multiply? Divide? Write

this below.

7. Solve the problem below.

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8. Check your answer.

9. Explain how you got your answer to your partner. Write what you said below.

10. Explain your answer to the rest of the group.

11. Write a similar problem on the back of this page.

I first modeled how to follow steps one through ten with a few problems so that

students knew what is expected of them. We discussed how using these steps would help

them solve math word problems. Once the participants understoond the process, I then

assigned students to partners and gave each set of partners a word problem to solve.

Students used the WPP to solve their word problem and discuss with the group how they

solved it.

Once students were using steps one through ten with more confidence, I

introduced the final step (Write a similar problem on the back of this page.) that asks

students to write their own word problem.

I scaffolded the process of writing word problems by using a word problem that

students had solved already using the WPP. The problem is as follows:

Sam’s truck weighs 4,725 pounds. The truck can carry 7,500 pounds of rocks.

What is the total weight of the truck and full load?

I then gave students a copy of the problem, leaving blanks for them to fill in the numbers

which were missing. The problem looked like this:

Sam’s truck weighs __________ pounds. The truck can carry _________ pounds

of rocks. What is the total weigh of the truck and full load?

33

Together as a group, we discussed changing the numbers and practiced solving the

problem with new sets of numbers. Finally, I gave them the following version of the

same problem:

Sam’s ______________ weighs __________ pounds. The _____________ can

carry _________ pounds of ______________. What is the total weigh of the

______________ and full load?

As a group, we discussed different vehicle choices and also options for different loads.

Students re-wrote the problem choosing one of the options we generated together.

Students took turns reading their new problems aloud to the group and we practiced

solving them, also together as a group.

Instruction continued this way with a few more problems, with partners of

students solving word problems using the WPP, and eventually writing their own version

of the problem with more and more information being provided by students.

Think-aloud. After completing instruction, I administered a think-aloud individually to

the participants. My goal was to see if the instruction was effective in increasing their

comprehension of math word problems. I recorded, transcribed, and then analyzed their

responses.

To analyze the think-aloud data, I used a checklist of strategies from the WPP to

code what students did and said during the think-aloud. My main goal was to see if

students were using the strategies they learned while using the WPP.

Checklist:

Circled words he/she didn’t understand / Discussed unfamiliar vocabulary

34

Discussed what the problem was asking them to find

Drew a picture to represent the problem

Discussed the function needed to solve the problem (addition, subtraction,

multiplication, division)

Solved the problem correctly

Checked his/her answer

Teacher journal. During the ten weeks of my study, I recorded my observations and

reflections in a journal. This provided anecdotal information about the process the

participants were going through in my study such as how they were using and reacting to

the WWP.

Post-test. At the end of instruction, I gave the participants a post-test similar to the one

they took before the instruction period began. I recorded these scores and compared them

to the pre-test scores, looking for improvement. The post-test format was the same as the

pre-test, in that I gave students five word problems to solve, and then the same problems

written in a number sentence form. See Appendix B for the post-test.

I compared each word problem with its number sentence counterpart and recorded

scores to reflect the number of word problems correctly answered and the number of

number sentence problems correctly answered. The five word problems I selected

represented at least one addition, one subtraction, one multiplication, and one division

problem to assess a variety of mathematical functions.

35

Materials Pre-test. I gave my participants a pre-test using selected word problems from the

Sample Fourth-Grade MCA Math Test (Minnesota Department of Education, 2009)

before I began instruction. The pre-test had two different forms; one form had word

problems, and the other form had number sentences that corresponded to the word

problems on the first form. See Appendix A for the pre-test.

Post-test. I administered a post-test using selected word problems from the Sample

Fourth-Grade MCA Math Test at the end of the ten-week instruction period. The post-test

also has two different forms; one form had word problems, and the other form had

number sentences that corresponded to the word problems on the first form. The two

forms of the post-test contain different problems than those found on the pre-test. See

Appendix B for the post-test.

Data Analysis The pre- and post-tests were scored to see how many items students answered

correctly. For both the pre-test and the post-test, I compared how students scored on each

word problem versus its number sentence counterpart. I also compared pre-test scores to

post-test scores to check for overall improvement. At the end of the ten-week instruction

period, I re-read my teacher journal and found common themes and insights. To analyze

the think-aloud data, I used a checklist of WPP strategies I taught students. For the

checklist, please refer to pages 33 – 34. My aim was to see if students were actually using

the steps they had learned in the procedure and if those steps were helpful to them when

solving a problem.

36

Verification of Data

To ensure the conclusions of this study were supported by sufficient data sources,

I have chosen to use quantitative as well as qualitative data collection methods. To

develop a baseline of students’ ability to solve math word problems, I administered a pre-

test and later a post-test to determine progress made after the instruction period. I also

recorded observations in a teacher journal and through that was able to compile insights

about how the students responded to and learned from the instruction. I also administered

think-alouds with individual students to see how students were applying the word

problem procedure to solve problems. It was by using all of these data collection methods

that I came to my final conclusions.

Ethics

This study employed the following safeguards to protect informant’s rights:

1. Parent permission was obtained before beginning my research.

2. Names of participants, school, and teachers were changed in my paper.

3. Students’ test scores and recorded think-aloud sessions and their transcriptions

will be destroyed six months after the study has been written up.

Conclusion In this chapter, I described the methods I used to carry out my research. I described my

use of a mixed method research paradigm. In addition, I gave information about the

participants, the setting, and my planned techniques for data collection and analysis.

Finally, I outlined the steps I took to make sure my research was done in an ethical

manner. In Chapter Four will I will present the results of my study.

37

CHAPTER FOUR: RESULTS

This chapter presents the results of the post-test, teacher journal, think-alouds, and

post-test of my study. I also include a discussion of how pre- and post-test scores

compared to each other. The data I collected helped to answer the overarching question

of how can I help ELLs to better comprehend and solve math word problems.

Specifically, I wanted to know how teachers could scaffold the process of teaching

students to write their own math word problems, if teaching students to write math word

problems would help them to comprehend word problems better, and if this process

would help students be able to solve math word problems with more accuracy. Lastly, I

wanted to investigate if teaching students to write such problems would help them to talk

about how they solve math word problems.

Pre-test Results

The participants in the study began by taking a pre-test before I began the

instruction focusing on solving math word problems. I gave participants a pre-test using

five selected word problems taken from the Sample Fourth-Grade MCA Math Test

(Minnesota Department of Education, 2009). The participants completed this written pre-

test during the small group intervention time. After collecting students’ papers, I then

gave them these same five problems written in number sentence form on a separate

paper. See Appendix A for the pre-test.

For all students, the scores for the word problem portion of the test were lower

than their scores for the corresponding number sentence portion. This tells me that

38

students were able to compute number sentence problems with more accuracy than they

were able to comprehend and solve word problems. For some students, like Student 1 and

Student 5, their ablitity to solve both the word problems and number sentences was fairly

accurate already. The other students had a more pronounced difference between their

scores for the word problems and number sentences. The participants’ pre-test scores can

be seen in table 4.1

Table 4.1

Student Pre-test Scores

Students Word Problem Score (Number correct out of 5)

Number Sentence Score (Number correct out of 5)

1 4 5 2 2 5 3 3 5 4 1 3 5 4 5 6 1 5

Teacher Journal Results During the ten weeks of instruction, I recorded my thoughts and observations

about how students were responding to the word problem instruction. After the

instruction period was completed, I looked back through my journal, searching for

common themes and insights in my comments. The themes I found included the

effectiveness of analyzing vocabulary and drawing pictures. Another common theme was

in noticing how difficult it was for students to explain verbally how they arrived at their

answers. Finally, I wrote about how pleased I was to see the students so engaged and

motivated while using the WPP and writing their own word problems.

39

One thought that I had involved analyzing the vocabulary in a word problem. In

one example I wrote about, we were working on the following problem:

9 children are coloring in class today. They have 3 dozen colored pencils. If they

divide the pencils up equally, how many pencils will each of them get?

While working through the WPP, I found out that a few students did not know the

meaning of the word dozen. Without this knowledge, it would have been impossible to

solve the problem. We talked about what dozen meant, giving examples of things that

come by the dozen like eggs, doughnuts, etc. The students were then able to proceed with

the problem and solve it correctly. It is really important for students to analyze the

vocabulary of word problems carefully. One word can make the difference in getting the

correct answer.

Another insight that I gained from my notes was that drawing a picture to

represent the word problem really seemed to help students to comprehend and solve it.

For example, during one class period students were working on the following problem:

Grandmother is teaching us how to sew. She has 30 yards of yarn, and she wants

to give each of us 4 yards. How many of us can she give yarn to, and how much

will she have left?

This was a very difficult problem for all of the students. There was a lot of discussion

about what the problem was asking us to find and how to solve it. It was not until we

drew a picture of the yarn and started cutting it up into sections of four yards each that

students were able to visualize the problem well enough to solve it. Over the weeks of

instruction, students used the strategy of drawing a picture to help them comprehend

40

different problems. It was a very effective tool for them. This was especially true of the

students who had lower word problem scores on the pre-test.

Another insight I had during this process was noticing how difficult it is for

students to explain verbally how they got their answers. I kept noting that we needed to

spend more time verbalizing mathematical thinking and processing. If I had had more

time in my study, this is one area that I would have expanded. Helping ELLs to practice

this is a really important way to help them develop their math as well as their oral

language skills.

One other common theme I wrote about in my journal was about how pleased I

was to see the students so engaged and motivated while using the WPP and writing their

own word problems. The WPP was such a helpful tool for students because it gave them

steps to follow which ultimately led to the successful solving of word problems. It was

especially helpful for the more emergent math students to have a formula to follow.

These were the students who, at the beginning of instruction, did not know where to start

when solving a word problem. The WPP gave them a path to successfully answering

questions. In addition, I noticed how engaged students were as they were writing their

own word problems. For once, they were not just handed a problem to solve. This was

different for them because they were able to create something that was theirs. For

example, we were working on a problem that read:

In our living room, we have 9 vases. There are 7 flowers in each vase. How many

flowers do we have in our living room?

41

After talking about other types of recepticles to replace the vase and other objects that

each recepticle could hold to replace the flowers, students wrote their own word

problems. They were really excited about being able to choose the options that they

wanted. One student chose to use 15 refrigerators and 12 cartons of ice cream. Another

student chose 20 baskets and 6 teddy bears. As each student came to the front of the

group to sit in the special chair to share their problem, I could tell that they were proud of

their problem and were really anxious for their peers to solve it. In fact, they tended to

choose more challenging numbers for their problems, which encouraged the math

calculations to get more difficult.

Think-aloud Results

I conducted think-alouds with the participants of the study. The think-aloud was

done with each student individually. Each student was given a paper with a math word

problem written at the top and a pencil. The problem was the following:

Mary went to the store and bought 7 boxes of crayons. Each box had 12 crayons

in it. How many crayons did Mary buy in all?

The instructions were to solve the word problem on the paper I provided as the

participant told me what he/she was thinking and doing while solving the word problem.

I began recording each participant and he/she solved the problem. I asked questions to

prompt him/her to explain what he/she was doing or explaining as she/he solved the

problem.

After all the recordings were complete, I transcribed the think-alouds. See

Appendix D for these transcriptions. Next, I coded the transcriptions using a checklist to

42

assess what students did/said. For this checklist, please refer to pages 33 – 34. I looked at

each student’s transcription individually, making tallies for each item from the checklist

that they had used. I made notes about each item for each student for further observations

about what students had said or done. Finally, I compared each student’s tallies and

notes, looking for patterns that I could use to summarize the information from all of the

think-alouds in general.

None of the students did the first item on the checklist (Circled words he/she

didn’t understand / Discussed unfamiliar vocabulary). There appeared to be no unfamiliar

words in the word problem that presented an issue for any of the participants.

Addressing item two on my checklist (Discussed what the problem is asking you

to find), all of the students had an understanding of what the problem was asking them to

find, but only two students (Students 3 and 5) explicitly stated the question that the

problem required them to answer. The other participants showed their understanding in

other ways like drawing a picture or ultimately finding the correct answer to the problem.

Student 3 was confused in attaching a label to her answer of 84, first saying “boxes” and

then changing her label to “crayons.” This suggests that her understanding of what the

problem was asking her to find perhaps was not as clear as some of the other students.

The third item on the checklist was “Drew a picture to represent the problem.”

Three students (Students 2, 4, and 6) drew a picture that consisted of seven boxes to

represent the boxes of crayons with either the number “12” or twelve tally marks to

represent the number of crayons in each box. One student (Student 3) wrote a column of

12s seven times to show that there were seven boxes of crayons with twelve crayons in

43

each box, but did not draw the boxes. The last two students (Students 1 and 5) did not

draw pictures, but instead directly went to writing the number sentence of 12 x 7.

It should be noted that the two students who did not draw pictures were the same

students who had high scores on the word problem portion of their pre-test. The other

students who either drew pictures or wrote 12 seven times had lower scores on the word

problem portion of the pre-test.

The fourth item on the checklist was “Discuss the function needed to solve the

problem (addition, subtraction, multiplication, division).” All students ultimately decided

that the function needed to solve their problem was either multiplication or repeated

addition, which is essentially the same as multiplication. Some students decided upon the

function and discussed their reasons why more confidently than others. Students 1, 2, and

5 were able to express the function they used confidently. Student 4 did not explain the

function she used, but showed that she needed to use repeated addition clearly in the

work she did on her paper. Students 3 and 6 were not confident about which function to

use to solve the problem. Student 3 first said she needed to multiply, and then said she

should add. When asked why, she said she thought she should add because the problem

used the words “in all.” Usually kids are taught these key words “in all” to mean they

should add. I believe that this information from past learning is what confused Student 3

in this case. I prompted her to use her paper to do whatever she needed to solve the

problem. It was at this point that she wrote seven 12s on her paper and solved the

problem using repeated addition. Student 6 first indicated that she should multiply and

then changed her mind, saying she needed to divide. I asked the student what she could

44

do on her paper to figure out what she needed to do. She was able to draw a picture and

find the correct solution to the problem. She was, however, still unable to verbalize the

mathematical function she had used at the end.

The sixth item on the checklist was “Solved the problem correctly.” All six

students were able to successfully solve the problem. Some were faster and more

confident than others, but they all arrived at the correct answer in the end.

The seventh checklist item was “Checked his/her answer.” None of the students

really did this step. They were all satisfied with their final answer once they had arrived

at it.

Post-test Results

The study ended with students taking a post-test after I had completed the ten-

week period of instruction focusing on solving math word problems. I gave participants

the post-test using five selected word problems taken from the Sample Fourth-Grade

MCA Math Test (Minnesota Department of Education, 2009).. The participants

completed this written post-test during the small group intervention time. After collecting

students’ papers, I then gave them these same five problems written in number sentence

form on a separate paper. See Appendix B for the post-test.

The participants’ post-test scores can be seen in Table 4.2. For all students, the

scores for the word problem portion of the test were the same as their scores for the

corresponding number sentence portion, with the exception of Student 3, whose word

problem score was 1 point lower than her number sentence score. These scores tell me

45

that students’ abilities to solve word problems were largely equivalent to their ability to

compute number sentence problems.

Table 4.2

Student Post-test Scores

Students Word Problem Score (Number correct out of 5)

Number Sentence Score (Number correct out of 5)

1 5 5 2 5 5 3 4 5 4 5 5 5 5 5 6 3 3

All participants in the study showed improvement in their scores on the word

problem portion of the tests. Results showing these improvements can be seen in Table

4.3. Student 1 improved by one point, Student 2 improved by three points, Student 3 by

one point, Student 4 by four points, Student 5 by one point, and Student 6 improved by

two points. The scores for the number sentence portion of the test largely stayed the

same, with the exception of Students 4 and 6 who each increased by two points.

I believe that the reason for the increases in the word problem test scores was

because of the time instructing students to use the WPP and the time closely analyzing,

pulling apart, and changing word problems as students wrote their own math word

problems. As students learned the WPP, it was necessary for them to solve each problem

in a methodical, step-by-step process. They needed to slow down and read problems

more carefully and identify the question the problem was asking them to solve. They

were then asked to draw a picture to represent what was happening in the problem. This

46

was especially helpful for those students with more emergent math skills to visualize the

problem. As students were working through these steps, I saw and heard them discussing

what problems meant and negotiating meaning with each other. The more emergent

students were often only able to determine the math function necessary to solve a

problem (mulitiplication, addition, etc.) after they had drawn a picture and discussed

what it meant. I saw that the partners’ careful analysis of problems led them to

successfully solve such problems.

When we started pulling apart and changing word problems, I began to notice

more confidence in students in comprehending the problems they were working with. As

we changed problems that students had solved previously, students were asked to

understand these problems more deeply than before. They not only had to interpret the

problems, but they also needed to think of different options for components in the

problems, making sure they still made sense. There were occasions when we were

brainstorming ideas together when new components suggested by students did not make

sense. The group discussed the ideas and helped to change them. For example, we were

working on the problem:

In our living room, we have 9 vases. There are 7 flowers in each vase. How many

flowers do we have in our living room?

Before students wrote their own version of the problem, we discussed options for the

components of vase and flowers together as a group. Someone suggested we change the

problem to how many boxes of ice cream were in a refrigerator. Another student said that

you wouldn’t find a refrigerator in the living room. We discussed as a group that perhaps

47

changing living room to kitchen would help this problem make more sense. This example

suggests that students were really thinking about the meaning of each problem as they

were making changes to it. It is this level of understanding that really helped students

comprehend the problem.

Because of these behaviors I saw as students worked through this process, I knew

that the instruction phase of this study was clearly time well spent for students to be able

to comprehend and solve these types of problems more accurately. I think that more time

and attention should be spent in the mainstream classroom working with math word

problems in the ways I have done in my research so that ELLs will be able to

comprehend and solve such problems.

Table 4.3

Pre-test and Post-test Comparison

Pre-test Post-test Student Word Problem

Score (Number correct

out of 5)

Number Sentence Score (Number correct

out of 5)

Student Word Problem Score

(Number correct out of 5)

Number Sentence Score (Number correct

out of 5) 1 4 5 1 5 5 2 2 5 2 5 5 3 3 5 3 4 5 4 1 3 4 5 5 5 4 5 5 5 5 6 1 5 6 3 3

48

Figure 4.1 Pre-test and Post-test Comparison Bar Graph WP Pre = Word Problem Pre-test score, WP Post = Word Problem Post-test score, NS

Pre = Number Sentence Pre-test score, NS Post = Number Sentence Post-test score.

0

1

2

3

4

5

6

1 2 3 4 5 6

Student

Scor

eWP PreWP PostNS PreNS Post

Chapter Five Overview

In Chapter Five I discuss my major findings, implications for teaching, limitations

of the study, professional growth and insight, and recommendations for further research.

49

CHAPTER FIVE: CONCLUSIONS

This chapter presents a discussion of my major findings, implications for

teaching, limitations of the study, professional growth and insight, and recommendations

for further research.

The purpose of my research was to explore how I could help ELLs to better

comprehend and solve math word problems. I wondered how scaffolding the writing of

word problems would help them to comprehend word problems better. I also wanted to

see if this process would help students be able to solve math word problems with more

accuracy. Lastly, I wanted to investigate if teaching students to write their own word

problems would help them to talk about how they solve math word problems.

Major Findings One major finding from the research was that, overall, the WPP was a helpful tool

for all students to use while solving math word problems, but different steps of the

procedure seemed to be most beneficial to students based on their level of math problem

solving ability. I found that the step which required students to draw pictures to represent

math word problems was a beneficial strategy for all students, but was especially useful

for the students that had lower scores on the word problem portion of the pre-test. I saw

these students that struggled in their ability to solve math word problems as the more

emergent students in the study based on their math ability. During the instruction phase

of the research, I made notes about this fact in my teacher journal. In talking with these

emergent students about how to solve problems, it was not until they drew a picture to

50

represent the problem that they were able to go on to successfully solve it. Also, I noticed

this strategy being particularly helpful during the think-alouds. The more advanced math

students were able to solve the think-aloud problem without drawing a picture. The other,

more emergent math students were unable to solve the problem until they drew a picture.

Another interesting finding from the research involves the more advanced math

problem solvers who scored quite high on the pre-test. At first, I was concerned that this

targeted instruction focusing on math word problems would not be as beneficial for them.

I realized as the instruction progressed that these more advanced students were also

benefiting from learning the WPP, most specifically the step which requires students to

explain their thinking about how they solved a problem. I noticed that the more advanced

students usually knew how to solve math word problems with relative ease, but had

difficulty explaining how they arrived at their answers. They would simply say, “I’m not

sure, I just knew the answer.” As we practiced using the WPP more and more, these

students improved their ability to explain the steps they took to solve the problem. This

oral language practice was beneficial to all the participants in the study.

The final step that asked students to write their own math word problem was

beneficial to all of the students. It required students to go beyond the mere reading and

solving of word problems, which is what they normally have to do. The dissection and

reconstruction of problems that students needed to do to change an existing problem led

students to really comprehend those problems more deeply. This was the step, as the

literature review indicated in Chapter Two, that really engaged students and got them

interested in working with word problems. Students were really excited about creating

51

problems that were their own. They were proud to share their problems and were also

very eager to solve each other’s problems. The level of engagement while creating,

sharing, and solving student authored math word problems was much higher than when

students were simply solving problems that were given to them to solve. As I also

mentioned in Chapter Two, when students write their own word problems, it can have a

positive effect on student motivation and interest. This activity links math to students’

personal experience and background knowledge, and allows students and teachers to

work collaboratively to create the math curriculum.

Discussion

My initial research question was how I could help ELLs to better comprehend and

solve math word problems. I feel that by performing this study I have made some real

progress toward answering that question. The instruction involving the teaching of the

Word Problem Procedure, including having students write their own word problems

suggests improvement in these students’ ability to comprehend and solve math word

problems.

For students who were already fairly strong in solving these types of problems,

their skills were simply reinforced. The most progress was seen in students who struggled

with comprehending and solving math word problems.

Looking at my more specific questions, I wanted to know how teachers could

scaffold the process of teaching students to write their own math word problems. I saw

through my research that slowly leading students through the process was an effective

way of scaffolding them. First, they changed numbers of existing problems. Later, they

52

changed both numbers and other details of a problem until they had their own unique

word problem that they could share and solve with their peers.

In another one of my specific research questions, I wanted to see if teaching

students to write math word problems would help them to comprehend word problems

better. By changing the word problems we had been working with and making the

problems their own, the students were able to have a deeper understanding of those

problems. Without this deeper understanding, they wouldn’t have been able to make the

changes that they did. It really made them take a deep look at what was happening in the

problems in a way that they wouldn’t have had to if they were simply solving them. They

had to go a few steps further. This helped them to further their understanding of the

problems and solve them with greater understanding and accuracy.

Another research question involved wanting to see if this process would help

students be able to solve math word problems with more accuracy. By conducting both a

pre-test and a post-test and comparing the scores, I was able to see improvement in all of

the word problem scores. This indicates to me that the instruction was successful in

helping the participants answer math word problems with more accuracy than they had

been able to do before the instruction period began.

In my final specific research question, I wanted to investigate if teaching students

to write their own word problems would help them to talk about how they solve math

word problems. I learned through the think-alouds that some students were confident in

describing the process they went through in order to solve a problem. Other students still

found this verbal explanation difficult. It was difficult to assess if there had been any

53

improvement without having done a think-aloud at the beginning of the instruction phase.

In retrospect, that would have given me a better understanding of where individual

students were in terms of explaining the thoughts they had while solving a math word

problem. Overall, through the course of the instruction, I noticed that most students found

it difficult to put into words that which they were thinking about while solving the

problems we were working on. I think that this is an area that all ELLs need to improve

upon. This should be considered when planning instruction in the mainstream classroom.

Time devoted to help ELLs practice oral language as they explain their thinking about

mathematical problems would be time well spent.

Implications for Teaching It is clear that teachers need to spend more time during math class teaching a

procedure that students can follow in order to solve word problems. Doing so would

seem to have great benefits for ELLs, as they are improving their language skills

alongside their math problem solving skills. With the rest of the math content teachers are

required to teach, there seems precious little time to devote to really delving deeply into

word problems. But in doing so, teachers can give their ELLs more chance for success in

really comprehending and solving such problems. This can only help them perform better

on math assessments that are largely made up of questions in word problem form.

I agree with the findings of Spanos’ 2003 research. Students need to learn some

kind of procedure in order to successfully solve math word problems. Especially for

ELLs, teaching this procedure through direct instruction helps them to break down

problems in a systematic way to aid in their comprehension. This means that teachers

54

would be well advised to spend the time teaching a procedure for their students to follow

to help them really understand math word problems.

I also think that the scaffolding process in teaching students to author their own

word problems is an important step for teachers to do. In past years, I have seen teachers

ask students to attempt to write their own word problems without scaffolding their work.

These problems often did not follow the same format that math word problems require,

and they often did not make a lot of sense. During my research, however, I saw all

students I was working with be successful in writing word problems. I attribute that

success to the scaffolding process I used that was recommended by Heinze (2005),

Spanos (1993), Winograd (1990, 1992), and Winograd and Higgins (1994/1995) in the

literature review. Therefore, teachers should scaffold the process by starting small and

working towards having students provide more and more information as they alter an

existing math word problem.

Limitations of the Study One limitation of the study was the amount of time I was able to devote to the

instruction of math word problems. I feel that I could have had a greater impact if I had

had more time to instruct students. In fact, researchers like Winograd advocate for the

writing and solving of mathematical problems every day (Winograd, 1992). I feel like the

research I conducted barely skimmed the surface largely due to time constraints.

Another limitation is the number of students who participated in my study. If I

had been able to include more than six students, I would have been able to collect more

data and I would have had a broader scope to my findings. Because I co-teach math in the

55

mainstream classroom, this setting would have helped to strengthen the work with word

problems we did. More students, both native and non-native speakers of English would

have provided a richer environment of experience and background knowledge in which to

explore, write and solve math word problems. ELLs in this scenario would have also had

access to more native English-speaking language models during the process than the

teacher.

Professional Growth and Insights

I feel that I have grown as a result of this experience, both as a teacher and a

researcher. Keeping a journal during the instruction phase of the research was especially

helpful in this sense. I was able to record and process my thoughts after each class, which

helped me become a more thoughtful and reflective teacher. It helped me to plan future

instruction better based on the needs of my students.

This research and planning the instruction to help students be more proficient in

solving math word problems has helped me to be better equipped in teaching math in the

future. As an ESL teacher, I have always felt more competent in teaching my students

language. As a result of my research, I now feel that I understand the language demands

of math better and that I am better able to help students navigate through these language

challenges. With Spanos’ Word Problem Procedure, I feel as though I can equip students

with the tools they need to comprehend and solve math word problems. I also feel that I

can help students engage more and become more motivated in solving word problems by

helping them to write their own word problems and share them with others.

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Further Research Recommendations

In my research I had only a limited amount of time to work with students and we

worked with a limited number of word problems. I would recommend a longer study that

could systematically look at each of the major mathematical functions that word

problems usually require: addition, subtraction, mulitiplication, and division. In addition

to learning the Word Problem Procedure, or some variation of it, and authoring word

problems, students could be asked to compare and contrast word problems that require

these four functions. They could compare language formulas that are used to have an

even deeper understanding of these different types of word problems.

In addition to conducting the study over a longer period of time, I would also

recommend doing the same research with more students in order to have a larger sample.

This larger group could consist of ELLs, or research could also be done in the

mainstream classroom. Teaching native and non-native English speakers the WPP would

give ELLs more native English-speaking models from whom to learn as students explain

their thinking and write and share their own math word problems. It would be interesting

research to be able to compare and contrast the process with this different class language

dynamic.

57

APPENDIX A

Pre-test questions

58

Pre-test Questions

Word problems

1. Last year a basketball player scored 513 points. This year he has scored 466 points.

How many more points must he score to have the same score as last year?

A. 47

B. 57

C. 153

D. 157

2. Jose, Adam, and Dan each bought 9 balloons. What is the total number of balloons the

3 friends bought?

A. 3

B. 12

C. 27

D. 36

3. Dana played a total of 28 soccer games during the months of June, July, and August.

She played 19 games in June and July. How many games did she play in August?

A. 8

B. 9

C. 10

D. 11

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4. Kelley made bead necklaces and sold them for $5 each. She earned $40. How many

necklaces did Kelley sell?

A. 6

B. 8

C. 35

D. 45

5. Gary read his book on 3 different weeks. 234 + 234 + 134 = __________

•On week 1, Gary read 234 pages of the book.

•On week 2, he read the same number of pages as on week 1.

•On week 3, he read 100 pages less than he read on week 2.

What was the total number of pages Gary read?

A. 334

B. 468

C. 568

D. 602

60

Corresponding number sentences

1. 513 – 466 = ___________

A. 47

B. 57

C. 153

D. 157

2. 3 x 9 = ___________

A. 3

B. 12

C. 27

D. 36

3. 28 – 19 = __________

A. 8

B. 9

C. 10

D. 11

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4. 40 ÷ 5 = __________

A. 6

B. 8

C. 35

D. 45

5. 234 + 234 + 134 = __________

A. 334

B. 468

C. 568

D. 602

62

APPENDIX B

Post-test questions

63

Post-test Questions

Word problems

1. There are 15 students waiting to have their school pictures taken. Each picture takes 3

minutes. What is the total number of minutes it will take for all of these students to have

their pictures taken?

A. 5

B. 12

C. 18

D. 45

2. A school had $1,000 to spend on a playground. They spent $651 on a new swing and a

slide. How much money does the school have left to spend?

A. $349

B. $451

C. $505

D. $651

3. A city has 3 schools.

•There are 633 students in elementary school.

•There are 587 students in middle school.

•There are 740 students in high school.

What is the total number of students in the 3 schools?

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A. 1,850

B. 1,860

C. 1,950

D. 1,960

4. Brad bought 5 sports drinks at the store. He spent a total of $10.00 on the drinks. How

much did each drink cost?

A. $1.50

B. $1.75

C. $2.00

D. $2.25

5. Pat and Tracy collected a total of 576 aluminum cans. Pat collected 398 aluminum

cans. How many did Tracy collect?

A. 178

B. 188

C. 222

D. 288

65

Corresponding number sentences

1.15 x 3 = __________

A. 5

B. 12

C. 18

D. 45

2. 1,000 – 651 = __________

A. $349

B. $451

C. $505

D. $651

3. 633 + 587 + 740 = ___________

A. 1,850

B. 1,860

C. 1,950

D. 1,960

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4. 10.00 ÷ 5.00 = __________

A. $1.50

B. $1.75

C. $2.00

D. $2.25

5. 576 – 398 = __________

A. 178

B. 188

C. 222

D. 288

67

APPENDIX C

Word Problem Procedure (WPP) sheet for students

68

Word Problem Procedure

1. Read the problem out loud.

2. Talk about the vocabulary and circle words you don’t understand. Write the words below.

3. Ask your partner or teacher for help with what these words mean.

4. What does the problem ask you to find? Write it below. (What question does it ask you?)

5. Draw a picture to represent the problem.

6. What should you do to solve the problem? Add? Subtract? Multiply? Divide? Write this below.

7. Solve the problem below.

8. Check your answer.

9. Explain how you got your answer to your partner. Write what you said below.

10. Explain your answer to the rest of the group.

11. Write a similar problem on the back of this page.

69

APPENDIX D

Think-aloud transcriptions

70

Student 1

Reads the problem

S: It’s telling me that there’s 7 boxes of crayons and there’s 12 crayons in each, so we do 12 times 7 and that equals, um…84. 84 crayons.

T: OK, good. So how did you know to multiply?

S: I know because it says in each box there are 12 crayons and there are 7 boxes. So, instead of just counting 12 seven times, I knew that was 12 times 7.

T: So, just the words helped you?

S: Yeah.

T: OK. Any other thoughts you had as you we solving the problem that helped you know what to do?

S: Well, really it was just easy because it was just 7 times 12.

T: OK, well and you are actually correct. Very nicely done.

Student 2

(reads the problem)

S: I think maybe to solve the problem maybe there’s a way to solve it. Maybe if you do the picture first. I’m doing this before I solve it.

(draws 7 boxes, each with the number 12 inside)

T: So it looks like you’re drawing some boxes. And it looks like the boxes have the number 12 inside. What do the boxes show? What do they represent?

S: They represent the crayons. So, I made 7 boxes. If I want to solve it I need to multiply.

T: OK.

S: I will use partial products.

T: Perfect.

(writes 7 x 12 and works out the problem)

71

S: And then the answer is 84 crayons.

T: Beautiful. Thank you.

Student 3 (reads the problem) T: Good. So, what do you think you should do to solve the problem? S: Add. T: OK. You can write anything you want to on the paper and tell me what you are doing. S: No, I think I should multiply. T: Why? S: I don’t know. T: Well, is there anything you could do on the paper that would help you know for sure? S: Because it says “in all.” T: Oh, so you think maybe you should add. Well, what do you think you should do? Go ahead and just show me. (writes down a column of seven 12s) T: It looks like you wrote down 12…how many times? S: Seven. T: How come? S: Because there’s 7 boxes and in each box there’s 12 crayons. T: Oh, so there’s 12 crayons in each of the 7 boxes so is that why you wrote 12 seven times? S: Yeah.

72

T: OK, and then what are you going to do? S: Add them. T: OK, sounds good. (adds the numbers) T: And what is the answer? S: She bought 84 boxes of crayons. T: 84 boxes of crayons? S: Yeah. T: Really? S: No, she bought 84 crayons. T: How do you know that? How do you know it’s crayons instead of boxes? S: Because it says here “How many crayons did Maria buy in all?” T: Uh huh. So your answer…your label would be… S: 84 crayons. T: OK, sounds good.

Student 4 (reads the problem) T: So, what do you think you should do to solve the problem? S: Make 7 boxes. T: OK, go for it.

(draws boxes)

T: OK, so it looks like you’re drawing 7 boxes.

73

S: And then I put 12 crayons in each box.

T: Alright, perfect.

(draws crayons)

T: OK, so it looks like you’ve got 12 in each box. Now what?

S: I’m gonna count them.

T: So, what is going on in your head?

S: I’m gonna add them…12 plus 12 plus 12…

T: Wow, so are you going to add all those 12s all in a row at one time?

S: I’m gonna add a few at a time.

T: Yeah, show me.

(student writes 12 + 12)

T: So, you’ve got 12 + 12. How much is that?

S: 24. And then 12 + 12 is another 24. 12 + 12 + 12 is 36.

T: Can you tell me how you did the adding there? I see on your paper 24 + 24 + 36. But, what was your first step?

S: I plussed 4 plus 4 plus 6 and that’s 14. And then I put the 1 over here and then I plussed 2 plus 2 plus 3 plus 1 and that’s 8.

T: OK, good. Then what?

S: Then I got 84.

T: OK, 84. And do you have to do another step to finish the problem?

S: Next, I have to label.

T: OK. So what’s your label?

S: Crayons.

T: OK, so is that your answer, 84 crayons?

S: Yes.

74

T: That looks beautiful and you got it right.

Student 5 (reads the problem)

S: The first thing I did was read the story. The second step is I’m gonna find the question that they’re asking me for.

T: Alright.

S: And I’m gonna underline it.

T: Yeah, do whatever you want and write whatever you want to show how to solve it.

S: So, if it’s asking me how many crayons did Mary buy in all, I have to multiply 12 times 7.

T: How did you know that?

S: Because it says that she bought 7 boxes and 12 crayons in each box, so that’s multiply. So, 12 times 7 equals…

T: So, it looks like you made some boxes there. What’s that about?

S: I’m doing partial products.

T: Perfect!

(works out the problem)

S: It’s 84 crayons.

T: How did you know it was crayons?

S: Because it says how many crayons.

T: You’re right, and that’s the right answer. Perfect.

75

Student 6

(reads the problem)

T: So, what do you think? How are you going to solve it? S: Multiply them? T: OK, why do you think multiply? S: Um, no, divide. T: How come? S: Because it said she bought 7 boxes and each box has 12 crayons. T: OK, is there anything you could do on the paper to help you to see what is happening in the problem? What could you do? S: Draw 7 boxes and each box has 12 crayons. T: OK, let’s see it. (draws picture) T: OK, it looks like you’ve got your 7 boxes with 12 tally marks in each box to show the 12 crayons. Now what? S: Now I have to count the crayons. T: OK, let’s see it. S: 84. T: OK, so you counted each crayon and you found there was a total of 84? S: Yes. T: So, then what? S: Then I put that I divided.

76

T: Is that what you did? You divided? You’re right she bought 84 crayons and you can see that in your picture. But did you divide numbers? Did you take a big number and divide it into smaller groups? S: No. T: What did you do? S: Multiply. T: You did. You multiplied. You’ve got 7 boxes, and that’s like 7 groups. And there are 12 in each group. So, 7 groups of 12 is the same as 7 times 12. And really, you can think about the problem as multiplication, or you can think about it like you just did it. You were adding. You added all of these numbers together. So really, you could say that it’s an addition problem. It’s repeated addition. You are repeatedly adding 12 + 12 + 12…(etc.) and then you get your answer of 84. Either way, if you’re multiplying 7 groups of 12 or adding 12 + 12 + 12…(etc.) you get the same answer. Can you write the answer down? (writes 84) T: And what would your label be? 84 what? (writes crayons) T: Good, so your answer is 84… S: Crayons. T: Good.

77

APPENDIX E

Parent consent letter – English

78

January 3, 2010 Dear Parent or Guardian, I am working on my Master’s in ESL (English as a Second Language) degree at Hamline University. To complete my degree, I need to do research in your child’s classroom. I will study how your child solves math word problems. I have permission from North Park and Hamline University to do this study. However, I also need to ask for your permission to have your child participate in my study. I already work with your child every day, teaching math and reading. Very soon, we will be focusing on solving math word problems and will even be writing some of our own math word problems. My study is designed to see how this teaching will help your child get better at solving math word problems. I will write a report about what I learn and it will be published and used to help other teachers. Your child will still learn about solving math word problems in class, even if I don’t include him/her in my report. Your child will not lose any learning time during this research study. I will not use your child’s name in my report or identify them in any way. All the information I collect about your child will be confidential. If you don’t want your child to be included in my report, that is OK. Also, your child can stop participating in the study at any time if you wish. If you have questions, please contact me at langenej@colheights.k12.mn.us (763-528-4298) or Ann Mabbott at Hamline University at amabbott@hamline.edu (651-523-2446). Thank you, Jennifer Langeness ELL Teacher, 4th Grade North Park Elementary School If you give permission for your child to be a part of this research study, please sign both letters. Return one letter to me and keep the other letter. My Child’s Name _______________________________ My Signature: _________________________________ Date: _______________________________________

79

APPENDIX F

Parent consent letter – Spanish

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03 de enero 2010 Estimado padre o tutor, Estoy trabajando en mi Maestría en ESL (Inglés como Segundo Idioma) en la Universidad Hamline. Para completar mis estudios, tengo que hacer la investigación en el salón de su hijo. Voy a estudiar cómo los niños resuelven problemas matemáticos. Tengo permiso de la escuela North Park y la Universidad de Hamline para hacer este estudio. Sin embargo, también necesito pedir su permiso para que su hijo participe en mi estudio. Ya estoy trabajando con su hijo todos los días, en las áreas de matemáticas y lectura. Muy pronto, nos enfocaremos en la solución de problemas matemáticos, e incluso vamos a crear nuestros propios problemas de este tipo. Mi estudio está diseñado para ver cómo ayudar a su hijo a mejorar en la resolución de problemas matemáticos. Voy a escribir un informe que se publicará y se utilizará para ayudar a otros maestros. Su hijo todavía aprenderá acerca de la solución de estos problemas en la clase, incluso si yo no lo incluyo su hijo en mi reporte. Su hijo no perderá el tiempo de aprendizaje durante este estudio de investigación. No voy a usar el nombre de su hijo en mi informe o identificarlo. Toda la información que recopilamos sobre su hijo será confidenciál. Si no quieres que su hijo sea incluido en mi informe, no hay problema. Además, su hijo puede dejar de participar en el estudio en cualquier momento si lo desea. Si tiene preguntas, por favor comuníquese conmigo al langenej@colheights.k12.mn.us (763-528-4298) o Ann Mabbott en la Universidad Hamline en amabbott@hamline.edu (651-523-2446). Gracias, Jennifer Langeness ESL Maestra, 4º Grado North Park Elementary School Si usted le da autorización a su hijo a ser parte de este estudio de investigación, por favor, firme ambas cartas. Regresar una carta dirigida a mí y mantener la otra carta para ustedes. Nombre del Niño/a _______________________________ Firma: _________________________________________ Fecha: _________________________________________

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REFERENCES Barwell, R. (2001). Difficulties with mathematical word problems. Proceedings of the British Society for Research into Learning Mathematics. Retrieved from http://www.bsrlm.org.uk/IPs/ip21-2/BSRLM-IP-21-2-11.pdf Barwell, R. (2003a). Patterns of attention in the interaction of a primary school mathematics student with English as an additional language. Educational Studies in Mathematics, 53, 35-59. Barwell, R. (2003b). Working on word problems. Mathematics Teaching, 185, 6-8. Basurto, I. (1999) Conditions of reading comprehension which facilitate word problems for second language learners. Reading Improvement, 36(3), 143-148. Chamot, A. & O’Malley, J.M. (1994). CALLA handbook: Implementing the cognitive academic language learning approach. MA: Addison-Wesley Publishing Company. Chamot, A., Dale, M., O’Malley, J.M., & Spanos, G. (1992). Learning and problem solving strategies of ESL students. Bilingual Research Journal 16(3&4) Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.20.9610&rep= rep1&type=pdf Chapman, O. (2006). Classroom practices for context of mathematics word problems. Educational Studies in Mathematics, 62, 211-230. Dörnyei, Z. (2007). Research methods in applied linguistics. Oxford: Oxford University Press. Heinze, K. (2005). The language of math. Presentation handouts from TESOL conference. Retrieved from www.rtmsd.org/74411258318268/lib/ 74411258318268/Language_of_Math.doc Hildebrand, C., Ludeman, C., & Mullin, J. (1999). Integrating mathematics with problem solving using the mathematician’s chair. Teaching Children Mathematics, 5(7), 431- 434.

Irujo, S. (2007). So just what is the academic language of mathematics? [Electronic version]. The ELL Outlook. Retrieved from http://www.coursecrafters.com/ ELL-Outlook/2007/may_jun/ELLOutlookITIArticle1.htm.

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Kang, H. & Pham, K. T. (1995) From 1 to Z: Integrating math and language learning. Paper presented at TESOL Convention (20th), Long Beach, CA. (ERIC Document Reproduction Service No. Ed 381 031). Martiniello, M. (2008). Language and the performance of ELLs in math word problems. Harvard Educational Review, 78(2), 333-368. McKay, S. (2006). Researching second language classrooms. New York: Routledge. Minnesota Department of Education (2009). Minnesota comprehensive assessments – series II: Mathematics item sampler, grade 4. Retrieved from http://www.education. state.mn.us/mdeprod/idcplg?IdcService=GET_FILE&dDocName=018176& RevisionSelectionMethod=latestReleased&Rendition=primary Moss, M. & Puma, M. (1995). Prospects: The congressionally mandated study of educational growth and opportunity. (First year report on language minority and limited English proficient students). Washington, DC: U.S. Department of Education. Retrieved from http://eric.ed.gov/ERICWebPortal/contentdelivery/servlet/ERIC Servlet?accno=ED394334 Short, D. & Spanos, G. (1989). Teaching mathematics to limited English proficient students. (ERIC Document Reproduction Service No. Ed 317 086). Spanos, G. (1993). ESL math and science for high school students: Two case studies. Paper presented at the Third National Research Symposium on Limited English Proficient Student Issues. Retrieved from http://www.ncela.gwu.edu/ pubs/symposia/third/spanos.htm. Rubenstein, R. N. & Thompson, D. R. (2002). Understanding and supporting children’s mathematical vocabulary development. Teaching Children Mathematics, 9(2), 107- 112. Winograd, K. (1990). Writing, solving, and sharing original math story problems: Case studies of fifth grade children’s cognitive behavior. Paper presented at the 1991 Annual Meeting of the American Educational Research Association. (ERIC Document Reproduction Service No. Ed 345 936). Winograd, K. (1992). What fifth graders learn when they write their own math problems. Educational Leadership, 49(7), 64-67. Winograd, K. & Higgins, K. (1994/1995). Writing, reading, and talking about mathematics: One interdisciplinary possibility. The Reading Teacher, 48(4), 310-318.